Multi-Cost function Genetic Algorithm (Multi-objective Genetic

In many control problems, there is more than one objective that needs to be optimised, the technique used to optimization these kind of problems is called Multi-objective optimization (Osyczka, 1985). The Multi-objective optimization problem is going to optimizes all the objective functions and normally those objective functions are conflicting or against each other. Such that improving one objective and reduces other objective. To optimize all objective function means to find out such a solution which will make one individual objective function optimal and have the other cost functions as optimal as possible. In multi-objective optimization problems, there is no single optimal solution, indeed there are family of optimization solutions. Because in multi-objective problems, the optimal solutions are the a set of compromise (or trade-offs) solutions (Edgeworth, 1881). Vilfredo Pareto used this idea to introduce the Edgeworth-Pareto optimum method or Pareto optimal (Pareto, 1896). In this method, a family of solutions are found which would improve one objective and at the same time does not make the other objectives worse. The set of Pareto optimal solution are called non-dominated solution, because those solutions are not dominated by other solutions. The plot of the non-dominated solution is called the Pareto front. In 1989, the Pareto-based technique called Vector Evaluated Genetic Algorithm (VEGA) suggested by Goldberg and Schaffer (Goldberg, 1989). This VEGA uses non-dominated ranking and selection to find out the Pareto front. They used the technique called “Fitness Sharing” to maintain the diversity of the population (Goldberg and Richardson, 1987). The fitness sharing technique will reduce the individual’s fitness if two individuals are similar. The Genetic algorithm based on the Pareto-front method is easy to implement, and does not involve choosing any

weighting factors which can be very difficult to choose in single cost function Genetic Algorithm. Moreover, the weight factor is even more difficult to choose if the objectives are in different variables and the multi-objective genetic algorithm avoids this problem. Therefore, the multi cost function GA has been used in this thesis. There is no straight minimum or maximum cost function in multi-objective genetic algorithm, but instead the domination method is used. All the best non-dominated solutions are Pareto-optimal solutions and include the fitness sharing method to maintain the diversity of the solution. Srinivas and Deb’s (Srinivas and Deb, 1995) has improved the single objective evolution algorithm to Nondominated Sorting Genetic Algorithm (NSGA). This algorithm needs to dominate the population again and again to find out the rank of each individual therefore this algorithm is very slow. Horn, Nafploitis and Goldberg (Horn et al, 1994) has improved the single objective genetic algorithm to Niched Pareto Genetic Algorithm (NPGA). This algorithm uses tournament selection based on Pareto dominance. This algorithm is fast but it has the chance to lose the local best. Because tournament selection is not going to select the best individual from the total population, but it selects the best individual from a fixed size individual, and those individuals are randomly chosen from total population. Therefore, there is a chance that the local best is missed. In 1995, Chipperfield A. and Fleming P. introduced the multi-objective genetic algorithm (MOGAs) into the design of a squared multivariable control system for a gas turbine engine (Chipperfield and Fleming, 1995). This multi-objective genetic algorithm method is evolving a family of Pareto-front solutions, these solutions allow the designer to examine the trade-off the individual objectives.

The University of Salford PHD Thesis

47 Yongwu Dong

For multi-objective GA, the Pareto optimal method is used. And the chart of this method is shown below:

In the Pareto front method, the rank of non-dominated is equal to the number of the result better than it. Which means the rank of non-dominated will be plus one if there is one other result which dominates it (Zitzler and Thiele, 1998). The rest of the procedure is similar to the single objective GA. Compared with the single objective GA, the multi objective GA optimises all the multi objective functions simultaneously. Therefore, there is no single best solution to be found, but a family of Pareto solutions exist. An individual solution belongs to the family of Pareto solutions as there is no other solution that can improve at least one of the objectives and improve another objective simultaneously (Horn et al, 1994).

Start Generation=generation+1 Is gen<max gen ? Sharing in current front Identify non-dominated individuals Create population Generation=0 Yes Stop

Figure 2.6: Flow chart of pareto front method No

In addition, as different optimisation problem requires the minimization or maximization, the Pareto front solution can be optimisation by minimization or maximization. If the problem involves minimization, the Pareto front has the form show below:

Figure 2.7: the concept of Pareto dominance.

As figure 2.7 shows, Point C is dominated by points A and B. Point A and B are better than point C both for objective 1 and objective 2. Point A does not dominate point B and point B does not dominate point A, because point A is the best point with respect to the objective 2 compare with point B and point B is the best one with respect to objective 1 compare with point A. In fact, points on the full line are not dominating each other. Hence all the points are located on the full line are non-dominated and possible optimal solutions, and they belong to the non-dominated Pareto solution (Kalyanmoy, 2002).

Objective 2 A B C Obje cti ve 1 Pareto Front

The University of Salford PHD Thesis

49 Yongwu Dong

If the Pareto front solution involves maximisation, the Pareto front has the form show below:

Figure 2.8: the concept of Pareto dominance.

As figure 2.8 shows, Point C is dominated by points A and B. Point A and B are better than point C both for objective 1 and objective 2. Point A does not dominate point B and point B does not dominate point A, because point A is the best point with respect to the objective 2 compare with point B and point B is the best one with respect to objective 1 compared with point A. In fact, points on the full line are not dominating each other. Hence all the points are located on the full line are non-dominated and efficient solutions, and they belong to Pareto solution (Kalyanmoy, 2002).